FINAL EXAM
Math 445, Term II, 2005-06
Due: See timetable on course website
Exam rules:
1. You may consult your textbook or the homework solutions or notes during the exam. You
are not allowed to discuss the exam with anyone besides the professor until the exams are
handed in. You may email me for an appointment or call me if you have a question during the
exam: 604 822 4973.
Office hours: April 20th,21st: 11-1
2. Answer all questions legibly. For credit, show all work.
3. You are required to sign the front page of the exam indicating that you have followed these
rules. Students found violating this pledge will fail the course.
4. Please attach this signed page to the front of your solutions.
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I have complied with the rules of this exam.
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1.(30pts)
2.(30pts)
3.(30pts)
3.(10pts)
Total(100pts)
1. Due to an increase in canine tooth decay, a pet supply store decides to sell toothpaste for
dogs. Following several months of data collection, they notice the following trends. At the usual
price of $3.20/tube, they sell 10 tubes/day. For every 20 cents that the price is discounted,
they sell 25% more toothpaste. Improvements to the label also increase sales. If they spend
5 cents per tube in improved labeling, the sales increase by 15%, with a maximum cost of 40
cents for an improved label. The store buys the tubes at 50 cents/tube, not counting any costs
for improving the label.
a)What is the optimal price and labeling investment for maximizing profits?
b)Test the sensitivity of the optimal price and labeling investment to the assumption that sales
increase by 15% for the label improvement.
c) What information does the Lagrange multiplier give about the profit?
2. In order to avoid unlimited growth of a cell population, it is believed that cells practice mitotic autoregulation, controlling growth by synthesizing a chemical that prevents mitosis from
taking place. The control chemical is known as the inhibitor. Let n denote the cell density
(number of cells/unit volume). In the absence of the inhibitor, the change in n is proportional
to cell density, governed by the rate coefficient α for population growth, and ω, the loss rate
coefficient for population decay. Let c be the concentration of the inhibitor (amount per unit
volume). Since c is produced by the cells, its growth is proportional to the cell density with
coefficient γ, with losses proportional to c, with loss coefficient δ. For c > 0, the growth rate
coefficent of the cell density is reduced by a factor 1/(1 + βc). For the following, use the parameter values: δ = 10, α = γ = 2, ω = β = 1.
a) Write down the continuous time model for cell density and concentration of the inhibitor.
b) Find all possible steady state solutions and sketch the vector field.
c) Determine the local stability of all the equilibrium points you found in part b), and draw
the complete phase portrait.
d) An approximate analytical solution can be found by assuming c is in steady state css .
With this assumption, solve for css in terms of n, and substitute into the equation for the cell
density to obtain a single variable model. How does the time behavior given by this model
compare with that of the full model you gave in part a)?
3. Consider a population composed of those susceptible to a disease, such as the flu, and those
infected with the disease. On any given day during the flu season (5 months), an infected
person can become susceptible again with probability 85%, or he or she can remain infected
with probability 15%. The susceptibles become infected with probability 30%, and and remain
susceptible with probability 70%.
a) Find the steady-state distribution of susceptibles and infecteds.
b) How do the model and the long-term behavior change if it is possible for people to join
a third group who recover from the disease and are no longer susceptible? Sketch the state
transition diagram, and describe the steady state behavior.
c) Suppose you take a sample of 1000 people which has the steady state distribution of susceptibles and infecteds which you found in part a). Give an estimate for the probability that more
than 290 are infected.
4. Meerschaert, Section 9.4, Exercise 9.
This exercise models the average waiting time for taxis, as discussed in class and summarized
as follows: a) Taxis arrive at a taxi stand with interarrival times which are exponentially distributed, with a mean of 5 minutes. You arrive at the taxi stand at exactly 1 hour. Simulate the
arrival of taxis using exponential random variables, and record the interarrival time between
the last taxi before 1 hour and the first taxi after 1 hour. Repeat this simulation 5000 times
and find the average of this particular interarrival time. Compare this to the result obtained
in class.
b) Repeat your computations from part a), but this time assume that the interarrival times of
the taxis are normal random variables, with the same mean and variance as in part a). How is
the “memoryless” effect evident in the difference between parts a) and b)?
There is an example code on the course website that you can modify. In this example code
the random variables are uniformly distributed. For parts a) and b), the matlab commands
exprnd and normrnd will be useful.
There are additional instructions on the course website if you are using a version of Matlab
which does not have these commands available.
Hand in a copy of the computer code which you used for parts a) and b) in this problem.
Include a description of what each line of the code does.